Proof Inequality: x ≤ y | Homework Statement

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Homework Statement



Let x and y be real numbers. Prove that if x =< y + k for every positive real number k, then x =< y

The Attempt at a Solution



x =< y + k
-y + x =< k
since k is positive, the lowest value it can take doesn't include 0: -y + x < 0
x < y

So I get x < y from x =< y + k and not the required x =< y. Am I right or I'm screwing up somewhere? Thanks for your help.
 
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If x and y are equal, then x \leq y + k for all k&gt;0 \in \mathbb R [/tex], right? So your conclusion must not be right.<br /> <br /> Try this. Assume there are an x and y such that x \leq y + k for all k&amp;gt;0 but x &amp;gt; y and show this leads to a contradiction.
 
Out of x > y I got -y + x > 0.
So that 0 < -y + x =< k

But that's not quite a contradiction? Or is it?
 
I haven't studied this at all, but intuitively I would say that lynchu is correct.

hgfalling said:
If x and y are equal, then x \leq y + k for all k&gt;0 \in \mathbb R [/tex], right?
<br /> <br /> If x and y are equal, then we only have x=y+k in the case that k=0, now if k&gt;0 then x&lt;y+k. Right?
 
Mentallic said:
I haven't studied this at all, but intuitively I would say that lynchu is correct.



If x and y are equal, then we only have x=y+k in the case that k=0, now if k>0 then x<y+k. Right?

Well yes, but that's not the statement that's being evaluated. The theorem to be proved is:

IF x \leq y + k for all real k&gt;0, THEN x \leq y.

lynchu's claim was stronger than this; he claimed that in fact

IF x \leq y + k for all real k&gt;0, THEN x &lt; y.

But this isn't true, because if x and y are equal, then the condition holds, but the conclusion is false.

What you said was:

IF x=y, THEN x&lt;y+k for all real k&gt;0.


lynchu said:
Out of x > y I got -y + x > 0.
So that 0 < -y + x =< k

But that's not quite a contradiction? Or is it?

No, not really. All you need to do is show me ONE k value so that if x \leq y + k and x &gt; y you get a contradiction.
 
Oh I see, yeah now it makes sense why it should be x\leq y :smile:
 
Appreciate the help so far.
I've spent way too much time on this and yet I just don't see it. :cry:
 
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